A Comprehensive Introduction to Differential Geometry, Volume 1, 2, 3rd Edition

655 pages | 1999 | DJVU | 8,5 Mb The "way forward" is Kozul's concept of the connection and this is introduced in Chapter 6. First, note that the connection here is one of the versions of the introduced by Kozul as a map of pairs of vector fields to a vector field. Another useful version, not studied in volume II, is to consider the connection as a Hessian which maps any smooth function to a bilinear form on the tangent space. Second, note that Chapter 6 is usually the starting point for most treatments of curvature in differential geometry (e.g Do Carmo's "Riemannian Geometry"). Without the motivating material from the previous chapters, it would be difficult to understand the need for(or the point of) Kozul's connection. Cartan's theory of curvature via a study of moving frames is detailed in Chapter 7. The author is careful to intuitively motivate Cartan's deviation from Euclidean concept as represented in the structure equations. Cartan's curvature tensor is shown to agree with Riemann's tensor, the "Test Case" is revisited, and the well-known fact that the curvature determines the Riemannian metric is established. Building on the orthonormal frames from the previous chapter, Spivak now considers Ehresmann's theory of connections in principal bundles in Chapter 8. The main results here introduce the Ehresmann connection on the frame bundle, and gives the Kozul connection as a Lie derivative, thought of as the Cartan connection obtained from the Ehresmann connection. My only complaint is that the author didn't include any exercises in this second volume. This is a real shame as the exercises in the first volume were very well-designed and one of the highlights of that text. Links (8,5 Mb) Quote:http://rapidshare.com/files/139503934/A_Comprehensive_Introduction_to_Differential_Geometry__Volume_1__3rd_Edition_www.softarchive.net.rar Quote:http://rapidshare.com/files/139503936/A_Comprehensive_Introduction_to_Differential_Geometry__Volume_2__3rd_Edition_www.softarchive.net.rar Reader Softwares (DJVU, PRC, LIT, PDF...): Quote:http://rapidshare.com/files/138511543/Reader_Softwares__www.softarchive.net.rar